I think the schools a kid goes to are really important.
My husband and I went around and around with the same idea - he firmly believed that a smart kid could succeed regardless of the elementary and high schools they attended. And, speaking from his experience, it worked out for him - he went to a public elementary school in a rural Mexican village with dirt floors and three grades in one classroom. His high school was a little better. He went on to graduate from a top university and has been successful in business. So, it worked for him. There were a lot of factors that contributed to it working for him, though - he had a college educated grandpa who supported him in all ways, he is really bright and he happened to be a part of a group of friends that all pursued going to the same top university.
After many, many conversations and visits to different types of schools as we searched both public and private schools (here in the US), he now agrees with me that while a kid can succeed in almost any school environment (with enough external support), it isn't always optimal. Once he went into some progressive and creative schools and saw the differences in teaching styles, materials and experiences available, expectations of the teachers and kids, interesting things other people are involved in and doing, etc, he was really surprised. He felt that while he did succeed in the environment he was in, he could have done so much more, and had a much broader experience and perspective, if he had been in a different environment.
I went to private schools almost my entire life, and am so thankful for the sacrifices my mom made to send me. I still feel it really made a difference in my life. The contrast between my private high school education and my brother's mostly public high school education was stark. He is a really smart guy, and went on to be successful academically in college. But, I definitely had the more enriched academic experience.
Ultimately, I think a kid can be successful coming from almost any school environment. If you can provide your kid with a more enriched school environment, that's great. I don't think it always boils down to public vs. private school - I have seen some amazing public schools, and both great and awful private schools. It is about finding the best fit for your child/ren, and not ruling out options only because they are private.
Good luck - I think it is a really hard topic to come to an agreement on.